[A]s an optimization
Bitcoin goes a step further and disallows invalid transactions to be published
in the blockchain at all, but that's the thing: that's just an optimization
that full-nodes don't actually need to operate.

Maybe I'm misunderstanding this, but it seems to be an interesting idea: the blockchain can contain invalid transactions, but all validating nodes will ignore them. I'm thinking, what if bitcoin was designed this way, or if someone created such hard fork, could it work? If yes, what drawbacks should be expected?

Correct me if I'm wrong, but probably Counterparty already works in this way, since they use bitcoin blockchain.

Last point is very solid, but I think bitcoin has or will have similar problem with fungibility: e.g. exchanges which don't want to share btc-e destiny will avoid accepting provably stolen money. OTOH, it doesn't actually mean the lack of consensus on blockchain.
– modularAug 5 '17 at 20:58

I believe Peter Todd was talking about a model where transaction validity is not part of the consensus rules, and nodes simply ignore transactions that are in violation, rather than rejecting the chain entirely.
– Pieter WuilleAug 5 '17 at 21:34